Lognormal process with Volatility(S,t)

johnywaker · June 21st, 2007, 1:13 pm

Can any one help me in valuing the first and second moment of an asset with the following Process:dS(t)=mu(t)*S(t)*dt +Sigma(S(t),t)*dWIn fact I'm just trying to value E[S(t)] and E[S(t)^2] Thanks,

LordR · June 21st, 2007, 1:22 pm

The first moment is as in Black-Scholes. The second moment will depend on your specification of sigma(S(t),t).

johnywaker · June 21st, 2007, 6:43 pm

Sigma(S(t),t) to me stands for the implied volatility surface.IS there any assumption that will make my life easier in valuing the second moment?If Sigma was just a function of t and not S(t), then I'll know how to value the second moment.Thanks

LordR · June 21st, 2007, 7:23 pm

d S^2 = 2SdS + Sigma(S,t)^2 dt = (mu * S^2 + Sigma(S,t)^2) dt + 2S * Sigma(S,t) dWWrite this in stochastic integral notation, take expectations, and if Sigma(S,t) is sufficientlyeasy you may be able to work it out.Hint: try Black-Scholes first.